# 2-Wasserstein (optimal transport) and extension to the set of all signed measures

Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as $$d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}$$ where the $\inf$ is over all couplings $\gamma$ of $\mu$ and $\nu$. Can we define a norm (or something norm-like) on the space of signed measures (or a linear subspace of it containing the cone of probability measures) which gives rise to $W_2$ for probability measures. (I suppose not, but why?)

If not, can we approximate $d_{W_2}$ by a norm?

• There is a dual formulation of Wasserstein distance which makes perfect sense for signed measures, although I don't know what pathological behaviors it might have in that generality. For comparison, this paper discusses the fact that if bounded-Lipschitz distance is extended in the obvious way to signed measures then it fails to be a complete metric: worldscientific.com/doi/abs/10.1142/S0219493712003584 – Mark Meckes Feb 12 '13 at 3:40
• Thanks for the reference. I will think more about the dual version. A more direct approach is also welcome. – passerby51 Feb 12 '13 at 4:21
• Related question: mathoverflow.net/questions/120291/… – Dirk Feb 12 '13 at 7:15

## 3 Answers

(I guess you missed a square in your definition.)

2-Wasserstein distance doesn't respect the convex structure on measures. Consider two points $x_1 \ne x_2$ and Dirac measures $\delta(x_1), \delta(x_2)$. The measure $\frac{\delta(x_1)+\delta(x_2)}{2}$ is not a midpoint between $\delta(x_1)$ and $\delta(x_2)$.

• You are right, I missed a square. I guess you are arguing why it can't be approximated by a norm? If I remember correctly, 2-Wasserstein is geodesically convex. So maybe it is possible to approximate it locally by a norm? – passerby51 Oct 25 '15 at 17:55

This paper has several links to relevant literature by Kantorovich & Rubinstein who define an OT inspired norm for signed measures.

https://hal.inria.fr/inria-00072186/en

For the Earth-mover $$W_1$$ distance (based upon the cost function $$c(x,y)=\|x-y\|$$) this is exactly the purpose of this paper. Note however that their construction does not work for the quadratic cost $$c(x,y)=\|x-y\|^2$$ as you seem to be hoping for.

• Maybe I'm misunderstanding something, but I think the fact that the $W_1$ distance extends to a norm on the space of all signed measures with finite first moment is actually an immediate consequence of the classical Kantorovich-Rubinstein theorem: namely, this theorem shows that the $W_1$-distance is just a restriction of the (metric induced by the) norm on the dual space of the space of Lipschitz continuous functions that vanish at infinity. I guess the paper you linked is up to something more involved. – Jochen Glueck May 24 at 10:56